# On the area of constrained polygonal linkages

## TL;DR

This paper investigates the configuration spaces of polygonal linkages with diagonal constraints, demonstrating that the oriented area functions as a Bott-Morse function and analyzing its critical points and indices.

## Contribution

It generalizes previous results on polygonal linkages to include more complex graphs like partial two-trees, providing a detailed analysis of the area function.

## Key findings

- Oriented area is a Bott-Morse function on the configuration space.
- Critical points of the area function are characterized.
- Bott-Morse indices are computed for these critical points.

## Abstract

We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are computed. This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova).